Comments 69 
a) The involutive and free operator \(dual\)  does not appear in the conclusions of Lemma REF . This suggests that this Lemma holds under more general conditions. However, some conditions are necessary.
Take for \(\mathfrak {C}\)  the collection of finite oriented graphs \(G\)  such that the symmetric hull of \(G\)  is a direct sum of finite paths. Let \(\varphi \)  be the map associating to each graph its symmetric hull. The class \(\mathfrak {C}\)  is a hereditary class and \(\varphi \)  is a free operator. Members of \(\mathfrak {C}\)  are prime iff their symmetric hull is prime, indeed their symmetric hull must have at most two vertices or must be a path. Thus the first conditions of the frame of our lemma are satisfied. The class \(\mathfrak {C}\)  is an age and the age of a prime structure. It is not minimal prime (it contains the age of an infinite zig-zag) but the range \(\varphi <\mathfrak {C}>\)  of \(\mathfrak {C}\)  being the age of the infinite path \(P_{\infty }\) , it is minimal prime. Hence \((3)\)  of Lemma REF  does not hold. In fact, \(\mathfrak {C}\)  contains \(2^{\aleph _0}\)  minimal prime ages, each one transformed to \(Age(P_{\infty })\)  by \(\varphi \) . For a proof of this fact, code orientations of \(P_{\infty }\)  by words \(u\)  with domain \({\mathbb {N}}\)  on the two letters alphabet \(\lbrace 0, 1\rbrace \) . To a 0-1 word \(u:= (u_n)_{n\in {\mathbb {N}}}\)  on \({\mathbb {N}}\)  associate an orientation of \(P_{\infty }\) , orienting the edge \(\lbrace n, n+1\rbrace \)  from \(n\)  to \(n+1\)  if \(u_n=1\)  and from \(n+1\)  from \(n\)  if \(u_n=0\) . Then observe that the age of the orientation is minimal prime if and only if the word \(u\)  is uniformly recurrent (a word \(u\)  on \({\mathbb {N}}\)  is uniformly recurrent if for every \(n\in {\mathbb {N}}\)  there exists \(m\in {\mathbb {N}}\)  such that each factor \(u(p),...,u(p+n)\)  of length \(n\)  occurs as a factor of every factor of length \(m\) ). To conclude, use the fact that there are \(2^{\aleph _0}\)  uniformly recurrent words \(u_\alpha \)  on the two letters alphabet \(\lbrace 0, 1\rbrace \)  such that for \(\alpha \ne \beta \)  the collections \(Fac(u_\alpha )\)  and \(Fac (u_\beta )\)  of their finite factors are distinct, and in fact incomparable with respect to set inclusion (this is a well known fact of symbolic dynamic, e.g. Sturmian words with different slopes will do [1]}). For more on the use of recurrent words, see [2]}.
